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Abstract
Structured intense laser beams offer degrees of freedom that are highly attractive for high-field science applications. However, the performance of high-power laser beams in these applications is often hindered by deviations from the desired spatiotemporal profile. This study reports the wavefront optimization of ultrafast Laguerre-Gaussian beams through the synergy of adaptive optics and genetic algorithm-guided feedback. The results indicate that the intensity fluctuations along the perimeter of the target ring-shaped profile can be reduced up to ∼15%. Furthermore, the radius of the ring beam profile can be tailored to a certain extent by establishing threshold fitting criteria. The versatility of this approach is experimentally demonstrated in conjunction with different focusing geometries.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the advent of chirped-pulse amplification (CPA) several decades ago [1,2], the interest in research relying on high-power lasers has been on a constant rise [3–6]. Most of the studies use Gaussian or super-Gaussian beams. Due to the complexity of the multi-stage amplification process accompanied by the cumulative effect of nonlinear processes and imperfections in the optics used in the laser system, the beam profile experiences a distortion in the form of wavefront aberrations. Low-order aberrations can usually be mitigated by the use of adaptive optics such as deformable mirrors (DM). However, many high-intensity laser applications would benefit from non-Gaussian spatiotemporal profiles that require the introduction of higher-order phase terms. With the experimental verification of orbital angular momentum (OAM) of light [7] and the advances in the beam phase shaping techniques, an exciting opportunity arose to explore the physical effects associated with spiraling Poynting vector [8]. Aside from being characterized by a helical wavefront, laser light carrying OAM exhibits a phase singularity at the beam center and an annular intensity profile. The azimuthal phase is characterized by the topological charge number ($m$). For instance, the azimuthal phase revolves one turn per wavelength in the forward direction when $m=1$. These beams are a subset of Laguerre-Gaussian modes with zero radial index. The radius of the optical vortex increases with the azimuthal index (topological charge), and its intensity decreases. The devices such as spatial light modulators (SLM) [9], spiral phase plates (SPP) [10], and q-plates [11] allow the development of more complex beam structures with relative ease. In contrast to the use of structured beams in telecommunications [12] or particle manipulation [13] that can take advantage of SLM versatility, high-intensity OAM-beam experiments require the use of high-damage threshold optics such as thin fused silica SPPs [10]. Recent advances in the development of off-axis spiral-phase mirrors [14] eliminate the dispersion effects on ultrafast pulses.
A growing body of theoretical advances [15,16] provides an insight into possible applications of high-intensity OAM laser beams. Further examples also include, but are not limited to, OAM transfer to a plasma wakefield [17,18], production of megagauss axial magnetic fields [19,20], laser-induced ion beam control [21], and the tailoring of laser wakefield topology to support the acceleration of electrons/positrons [22–24], to name a few. Ion acceleration experiments [25] demonstrated that OAM beams can reduce the emission solid angle and achieve higher proton energies in comparison to the Gaussian beam. Experiments further demonstrate realization of kilo-Tesla magnetic fields [26] that exhibit strong dependence on the laser beam waist.
The implementation of high-power OAM experiments is hindered by the nonlinearity of processes involved as well as the beam susceptibility to aberrations leading to profile distortions at focus. The influence of different types of aberrations on OAM beams has been recently explored in detail [27]. The authors report that Laguerre–Gaussian beams are highly sensitive to off-axis wavefront deformations, with astigmatism making the dominant contribution. They introduce ring contrast as the sensitive measure of beam quality. Previously mentioned examples rely on tight focusing, whereas another potential application of OAM beams, the multi-filamentation control [28,29], involves loose focusing. For instance, by controlling the arrangement of individual filaments through azimuthal fragmentation of the Laguerre-Gaussian beam, one could potentially generate a structure capable of supporting air waveguiding [30,31] over long distances. This capability may be suitable for enhancing the signal emission/collection in telecommunications [32,33] and remote sensing [34–39].
All mentioned applications would greatly benefit from improved beam structuring, which can be accomplished using adaptive optics. When it comes to mitigation of beam profile distortions, the automated control of adaptive optics is preferred due to the large number of parameters involved. This is typically achieved by dynamically shaping the DM via a feedback loop that employs a genetic algorithm (GA) [29,40–42]. This work demonstrates that the GA-controlled DM can optimize the wavefront of high-power ultrafast Laguerre–Gaussian beams in both loose and tight focusing configurations. The optimization process is based on a minimal variation of the intensity along the circle-fit of the beam profile (ring contrast). Depending on the number of data points included in the fit, the radius of the optimized beam profile can also be tailored. Several avenues for further improvements are discussed.
2. Experiment
A custom-built $\lambda ^3$ laser was used in the experiments, capable of delivering 800-nm pulses with an energy of up to 20 mJ and pulse duration of $\sim$40 fs at a repetition rate of 480 Hz. Figure 1 shows the experimental setup. Upon exiting the laser compressor, the beam is incident on the 47-mm wide DM (DM37PMNS4 by AOA Xinetics, Northrop Grumman) consisting of 37 piezoelectric actuators, spaced 7 mm apart in a square grid. The maximum displacement of each actuator is 4 µm. The GA controls the actuator voltages of the mirror while taking initialization settings of the camera (DMK 41BU02.H, Imaging Source) and user-defined fit as inputs. Calculation of the figure of merit (FOM) function is an integral part of the GA routine, described in more detail in Section 3. To measure the beam profile directly with the camera, the pulse energy was attenuated using two waveplate-polarizer pairs together with neutral density filters. The camera has a resolution of 1280 pixels $\times$ 960 pixels with 4.65-µm pixel width and 8-bit dynamic range. The helical wavefront of topological charge $m=1$ was produced employing 1-mm thick, 46-mm wide fused silica SPP. After reflecting off of the DM and passing through an SPP, the beam is focused onto a camera. The two focusing configurations used involve an f/2 off-axis parabolic mirror (OAP), and an f/100 lens. In the case of f/2 OAP, the beam was magnified using the 50$\times$ microscope objective and recorded with a camera.
Fig. 1. Simplified experimental scheme for LG$_{01}$ wavefront optimization in loose or tight focusing configuration (M - mirror, DM - deformable mirror, SPP - spiral phase plate, L - lens, OAP - off-axis parabolic mirror, MO - microscope objective).
3. Genetic algorithm optimization
Advanced optimization algorithms are often the tool of choice when the quantity of interest is a function of a large number of parameters. As an example, the so-called evolutionary algorithms refine the result using a random initial subset of the parameters involved and produce a solution by iteratively combining attributes of that subset. More specifically, GA is a class of evolutionary algorithms where the fitness of each candidate solution (child) can be expressed through a FOM. Each child consists of its attributes (genes). In this work, each mirror figure represents a child, and the position of each of the mirror actuators mimics the gene. Since they are based upon biological reproduction, GAs imply inheritance, crossover, and mutation of genes to create new children [43,44]. The user defines a starting mirror that can be random, a previously-optimized configuration, or a “flat” mirror of equal voltages on each deformable mirror actuator. The genetic algorithm evaluates a user-defined figure of merit to determine the fitness of each child - each set of actuator settings - and the goal of the algorithm in our case was to increase the FOM. The algorithm begins from the prescribed starting mirror setting and creates 10 initial parents, where each parent’s genes - each actuator’s voltage - are determined to probabilistically mutate by a random value from the starting gene based on a user-defined mutation percentage. The mutation percentage used in this work was 20%. The next - and each subsequent - generation creates 70 children from the 10 parents of the last generation, which are determined to be the 10 “most fit” individuals, i.e., the 10 individuals with the highest calculated FOM. By proceeding in this way, the genes which most improve the FOM have the highest probability to be passed on to future generations to achieve convergence towards the optimized mirror actuator voltage configuration. The optimum number of children, parents and the mutation percentage are user-defined and were found empirically for this specific experiment and could differ in other applications.
To illustrate an experimental scenario, Fig. 2 shows the SPP alignment procedure. Before introducing the SPP, the beam is first apodized with an iris diaphragm (a) to form a Gaussian-like profile. The center (singularity) of the SPP is translated such that its location approximately matches the centroid of the beam profile (b–d). The FOM used to determine the fitness of each child from the exit mode image is given by the min/max ratio [27] of pixel values along the perimeter of the circle used to fit the OAM-beam profile. The principle of identifying minimum and maximum values is illustrated in Fig. 3(a), with the least-squares circle fitting of the test profile. The intensity values of the test beam profile corresponding to the best fit are shown in Fig. 3(b). The FOM calculation routine is adjusted to select an arbitrary number of data points used for fitting by selecting an intensity-based threshold. This threshold is defined as Th = $X\times$Y$_i^{max}$, where the coefficient $X$ is the arbitrary real number that can take values in the range from 0 to unity, and $Y_i^{max}$ is the maximum gray-scale pixel value of the beam profile. For instance, if $X=0.5$ and $Y_i^{max}=240$, only the data points with pixel values higher than 120 will be considered for fitting. An apodized beam from Fig. 3 is illustrative of a simplified case, free from other distortions. The starting point for GA optimization is a non-apodized beam, as can be seen in more detail in Section 4. This methodology should also be applicable to other types of OAM-beam generation devices.
Fig. 3. (a) Least-squares circle fit of an experimental Laguerre-Gaussian beam profile; (b) normalized intensity distribution of pixel values corresponding to the circle fit.
4. Results and discussion
4.1 Loose focusing
The effects of modifying the FOM fitting coefficient can be seen in Fig. 4. The first column (a) presents circle fits (red line) of the non-apodized unoptimized beam arranged from $X=0.15$ to $X=0.6$ in the ascending order. The shape of the beam is elliptical and particularly distorted at its bottom-right edge. With increasing threshold, the number of data points used for circle fitting decreases, showing only the areas of the beam profile with a maximum intensity ($X=0.6$). Each of these test cases was subjected to 20 GA iterations (Fig. 4(b)), where the FOM exhibited a span from $\sim$0.1 to nearly 0.8. On our desktop computer with moderate processing power, the execution of the GA algorithm took about one minute per iteration. Multiple lines represent ten best mirror figures (children) in each iteration. During the first several iterations, the FOM progression is rapid in all cases and subsequently stagnates. The results of the optimization can be seen in Fig. 4(c). The ellipticity and distortion mentioned earlier are no longer apparent. However, it is also interesting to observe the effect of the threshold variation on the radius of the optimized beam profile, which can be seen primarily in the first three cases ($X=0.15$, 0.2, and 0.3). On one hand, with the threshold set to 0.15, the resulting profile seems most symmetric. On the other hand, $X=0.3$ not only results in a larger beam radius than for $X=0.15$, but also with less intensity variation along the perimeter defined by the fit. Figure 4(d) illustrates this with the intensity variation for both unoptimized (red line) and optimized (blue line) case. For $X=0.15$, the intensity varies about 20%, dropping to $\sim$15% when $X=0.3$. In other words, rejection of the data points corresponding below 30% of the maximum intensity has lead to a 5% improvement in beam uniformity compared to the scenario in which the threshold used for fitting is set to 15%. It should be noted that the initial intensity variation was greater when a greater number of data points was considered (Fig. 4(d), red line). As the number of data points available for circle fitting decreases, the same applies to the radius of the fitted circle. The region along the fit (Fig. 4(a)) having fewer data points is the source of intensity variation.
Fig. 4. Impact of the FOM-fitting variation on the: (a) fraction of the data points used in the unoptimized beam, (b) FOM progression within 20 GA iterations, (c) fraction of the data points used in the optimized beam, and (d) intensity variation along the profile perimeter for both unoptimized and optimized cases. The curves shown in column (d) are normalized to their corresponding mid-range. The red circle line is the best fit of the data points used ((a) and (c)).
Based on these findings, the fitting threshold variation could have two distinct uses: (i) intensity optimization and (i) uniformity optimization. We note that the former does not preclude latter and that a “compromise” between the two outcomes is also an option in this methodology ($X=0.2$). In loose focusing applications such as the ones relying on laser filamentation, improved uniformity can be advantageous. The unoptimized and optimized 3D profiles of the OAM-beam after 40 iterations and $X=0.3$ can be seen in Fig. 5. Similarly to the previous example, the $\sim$15% variation is achieved with relatively coarse actuator arrangement. A smoothed actuator displacement map of the DM after optimization is dominated by the strokes performed in horizontal direction (Fig. 5(b)). As a result, the burn profiles left on the surface of the compact disc can be seen in Fig. 6. The hot spots in the beam are indicative of nonlinear behavior and onset of filamentation. Figure 6(b) indicates the degree of regularity in azimuthal fragmentation in the form of 4 distinct lobes, which stands in contrast to the unoptimized case (6(a)). The ability to control the arrangement of the filaments may be beneficial for applications such as waveguiding in air [30,31] and/or discharge guiding [45–49].
Fig. 5. Optimization of Laguerre-Gaussian (LG$_{01}$) beam for f/100-lens focusing. (a) Unoptimized beam profile; (b) FOM progression and DM actuator displacement after 40 iterations; (c) optimized beam profile.
4.2 Tight focusing
In the case of tight focusing, the GA was run twice: first to optimize the Gaussian focal spot targeting the optimum wavefront without the SPP as in our previous work [40] and, subsequently, to perform ring-contrast-based optimization (Fig. 7). The intermediate step of aligning the SPP is analogous to the example shown in Fig. 2. Figure 7(a) shows an unoptimized beam profile, where two dominant lobes can be observed. The threshold for fitting was set to 0.08. Figure 7(b) shows that only one iteration increases FOM from about 0.1 to 0.3. However, after the first iteration, the FOM remains stagnant, resulting in a relatively modest reduction of the peak intensity variation ($\sim$30%, see Fig. 7(c)) when compared to the loose focusing case. We note that the mirror figure optimized for ring contrast is significantly different than the mirror figure optimized for wavefront. Both intensity modulations and phase modulations of the initial (non-optimized) beam profile would likely result in an azimuthally nonuniform ring structure; in the absence of optimization, an initially flat wavefront may not necessarily lead to the best possible ring profile. It should also be noted that the tight focusing configuration is more susceptible to variations in beam pointing stability. While the optimization results for both loose and tight focusing configurations are promising, further work is necessary to minimize the difficulties in the circle fitting procedure when tight focusing shot-to-shot variations in the beam pointing are significant. In other words, one solution to enhance the optimization performance in a tight focusing configuration is to improve the beam pointing stability. The second option might rely on developing an algorithm that would determine the dominant beam position while discarding the outlier locations. The latter approach could potentially avoid unnecessary strokes of DM actuators and improve the algorithm execution times.
Fig. 7. Optimization of Laguerre-Gaussian (LG$_{01}$) beam for f/2-OAP focusing. (a) Unoptimized beam profile with SPP; (b) FOM progression and DM actuator displacement for optimal wavefront (left inset) and after 8 iterations (right inset); (c) optimized beam profile.
5. Conclusion
In summary, GA optimization of Laguerre-Gaussian beam profiles of high-power pulses can enhance both the uniformity and intensity (profile radius). The time that the GA takes to perform such optimizations is on the order of tens of minutes using typical desktop computing infrastructure. For loose focusing (f/100), the optimization leads to nonuniformity of about 15%, while in the case of tight (f/2) focusing case the uniformity is about 30%. Additional work is needed to improve optimization performance in tight focusing, primarily in terms of better control of beam pointing and introducing other threshold criteria in the corresponding FOM. Modal decomposition will likely be helpful in these algorithm enhancements. Further improvements may also be expected by employing the DMs with a greater number of actuators. The present results should serve as motivation to the experimental high-field science community to further adopt deterministic evolutionary algorithm in the pursuit of unique capabilities provided by OAM-tailored beams.
Funding
Defense Threat Reduction Agency (HDTRA1-20-2-0002); Department of Energy National Nuclear Security Administration, Consortium for Monitoring, Technology, and Verification (DE-NA0003920).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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